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Chapter 1: Problem 50

Simplify by finding the absolute value. \(-|6-3|\)

Short Answer

Expert verified
-3

Step by step solution

01

- Calculate Inside the Absolute Value

First, determine the value inside the absolute value symbols. Subtract 3 from 6. The expression inside the absolute value becomes: \[ 6 - 3 = 3 \]
02

- Apply the Absolute Value

Next, find the absolute value of the result. The absolute value of 3 is: \[ |3| = 3 \]
03

- Apply the Negative Sign

Finally, apply the negative sign outside the absolute value. \(-|3| = -3 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

absolute value
The absolute value of a number tells us how far that number is from zero on the number line, without considering direction. It is always a non-negative number. For example, both \(3\) and \(-3\) are \(3\) units away from zero, so both have an absolute value of \(|3| = 3\).
To find the absolute value of a number, simply remove any negative sign in front of the number. In symbol form, this means \(|x|\) is equal to \(x\) if \(x\) is positive or zero, and equal to \(-x\) if \(x\) is negative.
Let's consider an example:
  • For \(|7|\), since \(7\) is positive, the absolute value is \(7\).
  • For \(|-5|\), since \(-5\) is negative, the absolute value is \(-(-5) = 5\).
Using absolute value helps us simplify expressions and focus on the magnitude, which can be very useful in solving various math problems.
integer subtraction
Integer subtraction involves finding the difference between two integers. This process can sometimes result in a positive number and sometimes a negative number.
In our example, we have \(6 - 3\). Here鈥檚 how you can think of this subtraction problem:
  • Visualize starting at \(6\) on a number line.
  • Since we are subtracting \(3\), move \(3\) units to the left.
  • We end up at \(3\).
This means \(6 - 3 = 3\).
To strengthen your understanding, here are a few more examples:
  • \(10 - 4 = 6\)
  • \(5 - 7 = -2\)
  • \(-3 - 2 = -5\)
Remember, subtraction can change the sign of a number, making it positive or negative depending on the values involved.
negative sign application
The negative sign indicates the opposite value of a number. It's essential in changing the sign of numbers, especially after calculating absolute values or when an expression requires negation.
In our example, once we find \(|3| = 3\), we then apply the negative sign that was originally in front of the absolute value. Here's how it works:
  • We start with \(|6 - 3|\) which simplifies to \(|3| = 3\).
  • Then, we apply the negative sign outside of the absolute value, leading to \(-|3| = -3\).
Other examples include:
  • \(-|8 - 2| = -|6| = -6\)
  • \(-|-4| = -4\)
Applying the negative sign is a straightforward but crucial step in simplifying mathematical expressions correctly.

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Most popular questions from this chapter

The two tables show the heights of some selected mountains and the depths of some selected trenches. Use the information given to answer. \( \)\begin{array}{|l|c|}\hline \text { Mountain } & {\text { Height (in feet) }} \\\ {\text { Foraker }} & {17,400} \\ {\text { Wilson }} & {14,246} \\\ {\text { Pikes Peak }} & {14,110} \\ \hline\end{array} $$ \begin{array}{|l|c|}\hline\text {} & {\text { Depth }( \text { in feet } } \\ {\text { Trench }} & {\text { as a negative number }} \\ {\text { Philippine }} & {-32,995} \\ {\text { Cayman }} & {-24,721} \\ {\text { Java }} & {-23,376} \\ \hline \end{array} $$ How much deeper is the Cayman Trench than the Java Trench?

Find each difference. $$ 3.4-(-8.2) $$

Perform each indicated operation. $$ -8-[(-4-1)+(9-2)] $$

A manufacturer has fixed costs of \(\$ 1000\) to produce widgets. Each widget costs \(\$ 5\) to make. The fixed cost to produce gadgets is \(\$ 750,\) and each gadget costs \(\$ 3\) to make. Write an expression for the cost to make \(x\) widgets. (Hint: The cost will be the sum of the fixed cost and the cost per item times the number of items.)

Find each difference. $$ 2-(3-5) $$
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